Internet memes are a powerful form of online communication, yet their nature and reliance on commonsense knowledge make toxicity detection challenging. Identifying key features for meme interpretation and understanding, is a crucial task. Previous work has been focused on some elements contributing to the meaning, such as the Textual dimension via OCR, the Visual dimension via object recognition, upper layers of meaning like the Emotional dimension, Toxicity detection via proxy variables, such as hate speech detection, and sentiment analysis. Nevertheless, there is still a lack of an overall architecture able to formally identify elements contributing to the meaning of a meme, and be used in the sense-making process. In this work, we present a semantic framework and a corresponding benchmark for automatic knowledge extraction from memes. First, we identify the necessary dimensions to understand and interpret a meme: Textual material, Visual material, Scene, Background Knowledge, Emotion, Semiotic Projection, Analogical Mapping, Overall Intent, Target Community, and Toxicity Assessment. Second, the framework guides a semi-automatic process of generating a benchmark with commonsense question-answer pairs about meme toxicity assessment and its underlying reason. The resulting benchmark M-QUEST consists of 609 question-answer pairs for 307 memes. Thirdly, we evaluate eight open-source large language models on their ability to correctly solve M-QUEST. Our results show that current models' commonsense reasoning capabilities for toxic meme interpretation vary depending on the dimension and architecture. Models with instruction tuning and reasoning capabilities significantly outperform the others, though pragmatic inference questions remain challenging. We release code, benchmark, and prompts to support future research intersecting multimodal content safety and commonsense reasoning.